There are some 'standard' applications of the adjoint functor
theorem (AFT) and the special adjoint functor theorem (SAFT), for
example, the existence of a free $\tau$-algebra (where
$\tau=$(operations,identities)) on a small set by the AFT,
Stone-Cech compactification by the SAFT, and, if I am not mistaken, the proof that the category of $\tau$-algebras is cocomplete (by
using the AFT to establish a left adjoint to the appropriate diagonal
functor).

However, I was not able to find any applications of the duals of
the AFT and the SAFT, neither in MacLane, nor in the Joy of
Cats.

The Joy of Cats contains the following intriguing
remark on p. 311:

Since many familiar categories have separators but
fail to have coseparators, the dual of the Special Adjoint Functor
Theorem is applicable even more often than the theorem itself.

But what are the mentioned application of the dual of the SAFT?

So my questions is: What are the 'standard' applications of the duals
of the AFT and the SAFT?

Googling for combinations of phrases like ''adjoint functor theorem''
and ''dual'' is not very useful, so I have tried ''dual of the adjoint
functor theorem'' and ''dual of the special adjoint functor theorem.''
This resulted in a total of 7 papers/books, from which I was not able
to get a clear answer to the current question. I have also tried in
The Wikipedia article on adjoint
functors, in nLab's
article on the adjoint functor
theorems, and
in someMO
questions, but without success.

1 Answer
1

One example is the construction of geometric morphisms. Any colimit-preserving functor between Grothendieck toposes has a right adjoint, so if it also preserves finite limits, then it is part of a geometric morphism. Of course, in many cases in practice, the right adjoint is also easy to write down explicitly.